to be a bifurcation point for periodic orbits is a straightforward con- sequence of such a continuation principle. Known results on open sets of euclidean spaces ...
PACIFIC JOURNAL OF MATHEMATICS Vol. 160, No. 2, 1993
A CONTINUATION PRINCIPLE FOR PERIODIC SOLUTIONS OF FORCED MOTION EQUATIONS ON MANIFOLDS AND APPLICATIONS TO BIFURCATION THEORY MASSIMO FURI A N D M A R I A PATRIZIA
PERA
We give a continuation principle for forced oscillations of second order differential equations on not necessarily compact diίferentiable manifolds. A topological sufficient condition for an equilibrium point to be a bifurcation point for periodic orbits is a straightforward consequence of such a continuation principle. Known results on open sets of euclidean spaces as well as a recent continuation principle for forced oscillations on compact manifolds with nonzero Euler-Poincare characteristic are also included as particular cases.
0. Introduction. Let M be a smooth (boundaryless) ra-dimensional manifold in W1 and consider on M a time dependent Γ-periodic tangent vector field, i.e. a continuous map / : R x M —> W1 with the property that, for all ( ί , ί ) € R x M , / ( / , ί ) is tangent to M at q and f(t + T, q) = f(t, q). The map / may be interpreted as a (periodic) force acting on a mass point q (of mass 1) constrained on M. A forced (or harmonic) oscillation on M is a Γ-periodic solution of the motion problem associated to the force / . In [FP4], in the attempt to solve the conjecture about the existence of forced oscillations for the spherical pendulum (i.e. for the case M = S2, the two dimensional sphere), we have studied the one-parameter motion problem associated to the force λf, λ > 0. In this context, we say that (λ, x) is a solution (pair) of the problem, if λ > 0 and x: R -> M is a forced oscillation corresponding to λf. Let us denote by X the set of all solution pairs. Since any point q e M is a rest point of the inertial problem (i.e. the motion problem with λ = 0), the constraint M may be regarded as a subset of X by means of the embedding q >-+ (0, q). With this in mind, we say that M is the manifold of trivial solutions of X and, consequently, any element of X\M will be a nontrivial solution (pair). We observe that in the nonflat case one may have nontrivial solutions even when λ = 0. Closed geodesies may be, in fact, Γ-periodic orbits if they have appropriate speed.
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MASSIMO FURI AND MARIA PATRIZIA PERA 1
If we consider the standard C metric structure on the space Cj(M) 1 of all Γ-periodic maps x: R —• M of class C , the main result in [FP4] can be stated as follows: THEOREM 0.1. Assume that the constraint M is compact with nonzero Euler-Poincare characteristic. Then X\M contains an unbounded connected subset whose closure in [0, oc) x Cγ(M) meets M.
In [FP4], an element q of the trivial subset M of X was called a bifurcation point (for the forced motion problem considered above) if any neighborhood of q contains a nontrivial solution, that is, if q is in the closure of X\M. So, as a consequence of the above result, one gets the existence of bifurcation points for a parametrized forced constrained system, provided that the constraint M is compact and χ(M) 9 the Euler-Poincare characteristic of M , is nonzero. We will show that a necessary condition for a point q G M to be a bifurcation point is that the average force / is zero at q i.e. T
Thus, Theorem 0.1 may also be regarded as an extension (or dynamical version) of the classical Poincare-Hopf theorem, which asserts that any tangent vector field on a compact boundaryless manifold M, with χ(M) Φ 0, vanishes somewhere. In fact, observe that a time independent tangent vector field / on M may be regarded as a periodic force (of any period). Theorem 0.1 above was successfully used in [FP5] to give an affirmative answer to the conjecture about the forced spherical pendulum. However, since the problem on whether or not any compact constrained system M with χ(M) Φ 0 has forced oscillations is still unsolved, we think that further investigations about one-parameter forced constrained systems may be of some interest. Our aim here is to extend to the noncompact case some of the results of [FP4], including the one above. The interest of this is mainly related to the fact that open sets in R m are noncompact manifolds. So, the above theorem does not apply to the flat case. Moreover, once the necessary condition for a point q e M to be a bifurcation point is fulfilled, the possibility of dealing with noncompact manifolds will give us the tools to restrict our attention to a neighborhood of q, in order to get sufficient conditions for bifurcation.
PERIODIC SOLUTIONS OF FORCED MOTION EQUATIONS
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In other words, what we do here is the spirit of [FP2] where our effort was devoted to first order differential equations on noncompact manifolds. n One may argue that a motion equation on a manifold M cR is just a special first order differential equation on the tangent bundle T(M)
n
n
= {(q, v) e R x R : q e M, v is tangent to M a t ? } .
However, a bifurcation problem associated to a force of the form λf, where, as above, λ > 0 and / : R x M —> Rn is a Γ-periodic tangent vector field on M, cannot be simply reduced to a problem of the form: z ( t ) = λg(t,z(t))9
ί€R,
z(t)eT(M),
studied in [FP2]. The reason is that the inertial motion problem does not correspond, in the phase space, to the trivial equation z{i) = 0. Actually, as is well known, the motion problem of a mass point q acted on by a force λf has the following form on T(M):
where the term h, which is a (nontrivial) tangent vector field on T(M), is related to the geometry of M and linear only in the flat case. This makes the problem in the non-flat case hard to deal with and cannot be handled with the techniques developed in [FP1] and in [Mar], where, for λ = 0, the problem was linear (with nontrivial kernel). We point out that a very interesting continuation principle for equations of the above form (and not necessarily related to second order equations) is given in [CMZ], where, roughly speaking, the equation is given on an open subset of a euclidean space and the existence of a branch of solution pairs (λ, z) is ensured provided that the Brouwer topological degree of h is (well defined and) nonzero. What seems peculiar to us, and interesting for further investigations, in our situation, is the role of g (or, equivalently, of the force /) which is important for the existence of a bifurcating branch. In fact, as we shall see, it is just the Euler characteristic of the average force / which, when defined and nonzero, ensures the existence of a global bifurcating branch. To see the relation with well-known concepts we recall that the Euler characteristic of a tangent vector field coincides with the Brouwer degree in the flat case and with the Euler-Poincare characteristic of the manifold in the compact boundaryless case.
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1. Notation and preliminaries. In this section we recall some definitions and results that will be needed in the sequel. The inner product of two vectors υ and w in Rn will be denoted by (v,w) and \υ\ will stand for the euclidean norm of υ (i.e. \v = (v,v){/2). Let M be an m-dimensional boundaryless smooth manifold in W and, for any q e M, let Tq(M) c W1 and T^M)1- c Rn denote respectively the tangent space and the normal space of M at q . Let T(M) denote the tangent bundle of M, i.e. the 2m-differentiable submanifold = {(q,v)eRnxRn:qeM,
veTq(M)}
of I " x P , containing a natural copy of M, via the embedding Given (
